<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="False" data-init="VerticalStudentView" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-global_warming_1" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="836e39be05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">Introduction</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@global_warming_text_gw_intro"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-block-type="html" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@global_warming_text_gw_intro" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="836e39be05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <p><b> Introduction</b></p><p> During this lesson we will introduce two simple models of the earth-atmosphere system, The Bare Rock Model and the Single Layer Model, to demonstrate how the greenhouse effect works. These simple models will help us to develop a basic understanding of the effect of solar radiation and atmospheric gasses on the temperature of Earth. First, we will develop the Bare Rock Model to calculate the temperature of the surface of the Earth with an atmosphere that is completely transparent to all incoming and outgoing radiation. In this model the predicted temperature will be lower then the observed average surface temperature of Earth. Then we will add to the Bare Rock Model a single layer of atmosphere that absorbs radiation emitted by Earth and calculate the temperature of the surface of Earth. In this model our calculation will be higher then the observed average surface temperature of Earth. A more detailed modeling of atmospheric physics is necessary in order to obtain a more accurate predicted value.</p><p> To build these models we need to introduce two new concepts: <i>radiative equilibrium</i> and <i>black body radiation</i> </p><p><i>Radiative equilibrium</i></p><center><img src="/assets/courseware/v1/7c74d717c3b7338c163943b94be5d8dc/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_gw_solarEarthRadiation2.svg" width="320"/></center><p> Earth absorbs part of the solar energy carried by the electromagnetic waves emitted by the sun. As a result, Earth heats up. In order to keep its temperature constant, Earth emits electromagnetic waves carrying energy back to space in such a way that the rate of energy emitted equals the rate of energy absorbed. The continuous process of absorption and emission creates a state of thermal equilibrium called <i>radiative equilibrium</i>.[1] </p><p><i>Electromagnetic radiation</i>: </p><p>Any object with a temperature above absolute zero kelvin emits electromagnetic radiation. For example, the red-hot electric burner of a stove or the coils of a toaster emit electromagnetic radiation that you can see. Even when the stove is at room temperature, the electric burner radiates electromagnetic energy. The difference is that the stove is emitting electromagnetic waves at frequencies below the infrared which the human eyes can’t detect. In fact, every object around you is emitting electromagnetic radiation all the time. </p><p><i>Black-body radiation</i>: </p><p> The sun and Earth both emit electromagnetic radiation. In order to quantify the emitted electromagnetic radiation, we introduce the concept of <i>blackbody radiation</i>. </p><p> A blackbody refers to an opaque object that emits thermal radiation. A perfect blackbody is one that absorbs all incoming light and does not reflect any. At room temperature, such an object would appear to be perfectly black (hence the term blackbody). If the blackbody is in thermal equilibrium with its surroundings, the emitted electromagnetic radiation is called <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html#c1">blackbody radiation</a>. [2] </p><p> The intensity \(I\) of the electromagnetic radiation emitted by a black-body depends on the temperature and is given by the <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/stefan2.html#c1">Stefan-Boltzmann law</a>: </p><p> \[I = \sigma T^4\] </p><p> where \(\sigma=5.67\times 10^{-8}\) Wm\(^{-2}\)K\(^{-4}\) is the Stefan-Boltzmann constant and \(T\) is the absolute temperature of the object in degrees Kelvin. </p><p><i>Temperature and Wavelength</i>:</p><p> The temperature of a blackbody not only determines the intensity of the electromagnetic radiation but also is related to the wavelength of the radiated electromagnetic waves. The larger the temperature of the object the shorter the wavelength of the emitted radiation. A plot of the intensity versus the wavelength of the emitted waves is called a <i>spectrum</i>.</p><p><center><img src="/assets/courseware/v1/5d32859479b69e427d5d65ceef1699b5/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_Black_body.svg" width="500"/></center></p><p>The shape of the curves depends on the temperature. Note that as the temperature goes up, the maximum of the curves gets higher. Also, the maximum of the curve occurs at a shorter wavelength. The relationship between the temperature and the wavelength for which the curve is a maximum is called the <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/wien.html#c2"><i>Wien's Displacement Law</i>:</a> </p> \[\lambda_{\mathrm{max}} = \frac{1}{4.965} \dfrac{hc}{kT} \] <p>where \(h = 6.63 \times 10^{-34} \mathrm{m^2 kg s^{-1}} \) is Planck's constant, \(c = 3.00 \times 10^8 \mathrm{m/s} \) is the speed of light, \(k=1.38\times10^{-23} \mathrm{kg m^2 s^{-2} K^{-1}} \) is Boltzmann's constant, and \(T\) is the temperature in kelvin. The law is often written as:</p> \[ \lambda_{\mathrm{max}} = \frac{b}{T} \] <p>where the constant of proportionality, \(b 2.90 \times 10^{-3} \mathrm{m K} \), is called <i>Wien's displacement current</i>.</p> <p> The temperature of the surface of the sun is about \(5778 \mathrm{K}\) and the intensity of the sun’s electromagnetic radiation has a maximum around wavelengths corresponding to visible light ( \(\lambda \approx 400 – 700\) nm). On the other hand, the average temperature of the surface of Earth is about \(300\) K and the intensity of the earth’s electromagnetic radiation has a maximum at wavelengths corresponding to infrared radiation (\(\lambda \approx 10 \;\mu\)m ) (Note: To show the Earth's spectrum, the curves in the figure below have different scales.) </p> <p><center><img src="/assets/courseware/v1/ac8f5745425bb82950a10d8611f4a107/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_Sun_Earth_spectrum.svg" width="600"/></center></p> <p>The wavelength of the electromagnetic radiation is important because the gases in the atmosphere absorb electromagnetic waves of certain wavelengths. For example, the ozone absorbs the ultraviolet solar radiation of wavelength \(\lambda \approx 200 – 300\) nm . This is fortunate because ultraviolet radiation is highly damaging to living organisms. On the other hand, water vapor and carbon dioxide absorb the infrared radiation emitted by the Earth. These gasses are some of the main contributors to the greenhouse effect. </p> <p><b> Notes and References:</b></p> <p>[1] Radiative equilibrium refers to a thermal equilibrium in which radiation is the only mean by which the object loses energy. In general, heat can also flow by conduction and convection, or by work done on or by the system. Thus radiative equilibrium applies when the object is not doing work and conduction and convection losses are small or absent. </p> <p>[2] Feynman Lectures on Physics, Volume 1, Chapter 41</p> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="False" data-init="VerticalStudentView" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-global_warming_2" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="836e39be05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">Problem 1</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_01"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_01" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="836e39be05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_gw_gw_01" class="problems-wrapper" role="group" aria-labelledby="gw_gw_01-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_01" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_01/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="2" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;gw_gw_01-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_01-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 1: The Solar Power at the top of Earth&amp;#39;s Atmosphere &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_01-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; The goal of this problem is to relate the time averaged power of the electromagnetic energy radiated by the sun, [mathjaxinline]P_{s}[/mathjaxinline], and the intensity of the electromagnetic waves at the top of Earth&#39;s atmosphere, known as the solar constant, [mathjaxinline]S_0[/mathjaxinline], which is the time average of the Poynting vector at the top of Earth&#39;s atmosphere. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; Assume that the sun radiates electromagnetic waves uniformly in all directions as if it were a point source, calculate [mathjaxinline]S_0[/mathjaxinline], the intensity of the electromagnetic waves at the top of Earth&#39;s atmosphere in terms of P_s for [mathjaxinline]P_ s[/mathjaxinline], and R_0 for [mathjaxinline]R_0[/mathjaxinline], the mean radius of Earth&#39;s orbit around the Sun. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]S_0=[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_gw_gw_01_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_gw_gw_01_2_1&#34; id=&#34;input_gw_gw_01_2_1&#34; aria-describedby=&#34;status_gw_gw_01_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_gw_gw_01_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_gw_gw_01_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_gw_gw_01_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_gw_gw_01_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_gw_gw_01_2_1_dynamath&#34; name=&#34;input_gw_gw_01_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_gw_gw_01_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part b)&lt;/b&gt; The averaged measured value is [mathjaxinline]S_0 = 1.36 \; \mbox{kW/m}^2[/mathjaxinline] [1]. The mean radius of the Earth&#39;s orbit around the sun is [mathjaxinline]R_{0} = 1.50\times 10^8 \; \mbox{km}[/mathjaxinline], the averaged power emitted by the sun is: &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_gw_gw_01_3_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_gw_gw_01_3_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_gw_gw_01_3_1&#34; id=&#34;input_gw_gw_01_3_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;gw_gw_01_3_1-choice_1-label&#34; for=&#34;input_gw_gw_01_3_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_gw_gw_01_3_1&#34;&gt; &lt;text&gt; [mathjaxinline]3.8\times 10^{20} \; \mbox{W}[/mathjaxinline]&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_gw_gw_01_3_1&#34; id=&#34;input_gw_gw_01_3_1_choice_2&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;gw_gw_01_3_1-choice_2-label&#34; for=&#34;input_gw_gw_01_3_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_gw_gw_01_3_1&#34;&gt; &lt;text&gt; [mathjaxinline]3.8\times 10^{26} \; \mbox{W}[/mathjaxinline]&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_gw_gw_01_3_1&#34; id=&#34;input_gw_gw_01_3_1_choice_3&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;gw_gw_01_3_1-choice_3-label&#34; for=&#34;input_gw_gw_01_3_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_gw_gw_01_3_1&#34;&gt; &lt;text&gt; [mathjaxinline]3.8\times 10^{17} \; \mbox{W}[/mathjaxinline]&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_gw_gw_01_3_1&#34; id=&#34;input_gw_gw_01_3_1_choice_4&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_4&#34;/&gt;&lt;label id=&#34;gw_gw_01_3_1-choice_4-label&#34; for=&#34;input_gw_gw_01_3_1_choice_4&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_gw_gw_01_3_1&#34;&gt; &lt;text&gt; [mathjaxinline]3.8\times 10^{36} \; \mbox{W}[/mathjaxinline]&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_gw_gw_01_3_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_gw_gw_01_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_gw_gw_01_solution_2&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 1: The Solar Power at the top of Earth&amp;#39;s Atmosphere &#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_gw_gw_01&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_gw_gw_01&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;gw_gw_01-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;gw_gw_01-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;gw_gw_01-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@global_warming_text_gw_01_ref"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-block-type="html" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@global_warming_text_gw_01_ref" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="836e39be05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <p>[1] References:</p><p><a href="https://science.nasa.gov/science-news/science-at-nasa/2003/17jan_solcon">a. Measuring the Solar Constant (Solar Constant Radiometer).</a></p><p><a href="http://lasp.colorado.edu/home/sorce/2016/12/05/new-climate-records-released-are-based-on-sorce-measurements/">b. Measurements of the solar constant, or Total Solar Irradiance (SORCE - Solar Radiation and Climate Experiment).</a></p> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="False" data-init="VerticalStudentView" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-global_warming_3" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="836e39be05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">Problem 2</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_02"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_02" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="836e39be05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_gw_gw_02" class="problems-wrapper" role="group" aria-labelledby="gw_gw_02-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_02" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_02/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="3" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;gw_gw_02-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_02-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 2: Averaged Incident Power at the top of Earth&amp;#39;s atmsophere &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_02-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; Average incident solar power at the top of Earth&#39;s atmosphere: &lt;/p&gt; &lt;p&gt; In the previous problem we considered the sun to be a source of electromagnetic waves radiating uniformly in all directions. Because the radius of Earth is much smaller than the mean radius of Earth&#39;s orbit around the sun ([mathjaxinline]R_ E=6.37 \times 10^3\; \mbox{km} &amp;lt;&amp;lt; R_{0} = 1.5 \times 10^8 \; \mbox{km}[/mathjaxinline]), the electromagnetic waves emitted by the sun can be assumed to be planar by the time they reach Earth. &lt;/p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/a141c35ee8f9d7f2a87917f789fce736/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_gw_03.svg&#34; width=&#34;440&#34;/&gt; &lt;/center&gt; &lt;p&gt; Calculate [mathjaxinline]P_{top}[/mathjaxinline], the time averaged power of solar radiation at the top of Earth&#39;s atmosphere. Express your answer in terms of S_0 for [mathjaxinline]S_0[/mathjaxinline], and R_E for [mathjaxinline]R_ E[/mathjaxinline]. (You can neglect the depth of the atmosphere of [mathjaxinline]30[/mathjaxinline] km with respect to the radius of Earth, [mathjaxinline]R_ E = 6.37 \times 10^3[/mathjaxinline] km.) &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]P_{top} =[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_gw_gw_02_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_gw_gw_02_2_1&#34; id=&#34;input_gw_gw_02_2_1&#34; aria-describedby=&#34;status_gw_gw_02_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_gw_gw_02_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_gw_gw_02_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_gw_gw_02_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_gw_gw_02_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_gw_gw_02_2_1_dynamath&#34; name=&#34;input_gw_gw_02_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_gw_gw_02_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part b)&lt;/b&gt; Incoming average solar power absorbed by Earth. &lt;/p&gt; &lt;p&gt; When the electromagnetic radiation from the sun reaches the top of Earth&#39;s atmosphere part of it is reflected back into space and the other part is absorbed by Earth. The fraction of energy reflected by a particular surface that arise from a variety of effects is called the &lt;i class=&#34;itshape&#34;&gt;albedo&lt;/i&gt; and it is denoted by the letter [mathjaxinline]\alpha[/mathjaxinline]. The yearly average albedo of Earth is within a few percent [mathjaxinline]\alpha = 0.30[/mathjaxinline] ([1]). Calculate, [mathjaxinline]P_{in}[/mathjaxinline], the time averaged rate of incoming solar radiation absorbed by Earth in terms of alpha for [mathjaxinline]\alpha[/mathjaxinline], S_0 for [mathjaxinline]S_0[/mathjaxinline], and R_E for [mathjaxinline]R_ E[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]P_{in}=[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_gw_gw_02_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_gw_gw_02_3_1&#34; id=&#34;input_gw_gw_02_3_1&#34; aria-describedby=&#34;status_gw_gw_02_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_gw_gw_02_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_gw_gw_02_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_gw_gw_02_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_gw_gw_02_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_gw_gw_02_3_1_dynamath&#34; name=&#34;input_gw_gw_02_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_gw_gw_02_solution_2&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part c)&lt;/b&gt; Incoming intensity of the solar radiation absorbed by Earth. &lt;/p&gt; &lt;p&gt; From our experience, we know that the daytime side of Earth is warmer than the nighttime side, and it is even warmer at the equator. In our simple model, we will assume Earth is a uniform spherical blackbody that perfectly absorbs incoming solar radiation and radiates it back into space at a temperature [mathjaxinline]T_ E[/mathjaxinline]. Under this assumption, calculate [mathjaxinline]I_{in}[/mathjaxinline], the intensity of the incoming solar radiation absorbed by Earth. Express your answer in terms of alpha for [mathjaxinline]\alpha[/mathjaxinline] and S_0 for [mathjaxinline]S_0[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]I_{in}=[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_gw_gw_02_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_gw_gw_02_4_1&#34; id=&#34;input_gw_gw_02_4_1&#34; aria-describedby=&#34;status_gw_gw_02_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_gw_gw_02_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_gw_gw_02_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_gw_gw_02_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_gw_gw_02_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_gw_gw_02_4_1_dynamath&#34; name=&#34;input_gw_gw_02_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_gw_gw_02_solution_3&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt; [1] The albedo varies with the amount of coverage of clouds, ice, and snow. On average, Earth&#39;s albedo is 0.30: for clear sky it is [mathjaxinline]\alpha \approx 0.15[/mathjaxinline], when fully covered by clouds or snow [mathjaxinline]\alpha \approx 0.60[/mathjaxinline]. It also depends on the particles suspended in the atmosphere. For example,the albedo of Venus is [mathjaxinline]\alpha \approx 0.70[/mathjaxinline] due to the thick layer of sulfuric acid in its atmosphere - that is the reason why it appears so bright in the sky. &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 2: Averaged Incident Power at the top of Earth&amp;#39;s atmsophere &#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_gw_gw_02&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_gw_gw_02&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="False" data-init="VerticalStudentView" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-global_warming_4" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="836e39be05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">Problem 3</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_03"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_03" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="836e39be05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_gw_gw_03" class="problems-wrapper" role="group" aria-labelledby="gw_gw_03-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_03" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_03/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="2" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;gw_gw_03-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_03-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 3: The Bare Rock Model &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_03-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Earth absorbs the incoming solar radiation and it warms up. In order to keep the same temperature [mathjaxinline]T_ E[/mathjaxinline] at all times, Earth must be in &lt;i class=&#34;itshape&#34;&gt;radiative equilibrium&lt;/i&gt; with its surroundings. Therefore, &lt;/p&gt; &lt;center&gt;&lt;i class=&#34;itshape&#34;&gt;The solar radiation energy absorbed by Earth must equal the radiation emitted by Earth&lt;/i&gt;. &lt;/center&gt; &lt;p&gt; We will first make the assumption that the radiation emitted by Earth does not interact with the gasses of the atmosphere, Earth&#39;s radiation escapes out to space as if there were no atmosphere. In addition, we assume that Earth radiates as a perfect spherical blackbody. This model is called the &lt;i class=&#34;itshape&#34;&gt;Bare Rock Model&lt;/i&gt;. &lt;/p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/96aff7d3a80abda507d9ad467eb3ee2d/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_gw_03_2.svg&#34; width=&#34;550&#34;/&gt; &lt;/center&gt; &lt;p&gt; As a result, the intensity of the energy radiated by Earth can be calculated using the Stefan-Boltzmann law: &lt;/p&gt; &lt;p&gt; [mathjaxinline]I_{out} = \sigma T_ E^4[/mathjaxinline] &lt;/p&gt; &lt;p&gt; where [mathjaxinline]\sigma[/mathjaxinline] is the Stefan-Boltzmann constant, and [mathjaxinline]T_ E[/mathjaxinline] is the temperature of the surface of Earth. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; Using the expression for the intensity of the solar radiation absoprbed by Earth obtained in Problem 2, [mathjaxinline]\displaystyle I_{in}= \frac{1-\alpha }{4}S_0[/mathjaxinline], calculate temperature of the surface of Earth, [mathjaxinline]T_ E[/mathjaxinline] in terms of S_0 for [mathjaxinline]S_0[/mathjaxinline], alpha for [mathjaxinline]\alpha[/mathjaxinline], and sigma for [mathjaxinline]\sigma[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]T_ E=[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_gw_gw_03_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_gw_gw_03_2_1&#34; id=&#34;input_gw_gw_03_2_1&#34; aria-describedby=&#34;status_gw_gw_03_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_gw_gw_03_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_gw_gw_03_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_gw_gw_03_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_gw_gw_03_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_gw_gw_03_2_1_dynamath&#34; name=&#34;input_gw_gw_03_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_gw_gw_03_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part b)&lt;/b&gt; Using the numerical values of the solar constant [mathjaxinline]S_0=1.36[/mathjaxinline] kW/m[mathjaxinline]^2[/mathjaxinline], Earth&#39;s albedo [mathjaxinline]\alpha = 0.30[/mathjaxinline], and the Stefan-Boltzmann constant [mathjaxinline]\sigma = 5.67\times 10^{-8}[/mathjaxinline] W/m[mathjaxinline]^2[/mathjaxinline]K[mathjaxinline]^4[/mathjaxinline], calculate the temperature of the surface of Earth. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]T_ E=[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_gw_gw_03_3_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_gw_gw_03_3_1&#34; id=&#34;input_gw_gw_03_3_1&#34; aria-describedby=&#34;trailing_text_gw_gw_03_3_1 status_gw_gw_03_3_1&#34; value=&#34;&#34; size=&#34;10&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_gw_gw_03_3_1&#34;&gt;[mathjaxinline]\mathrm{(K)}[/mathjaxinline]&lt;/span&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_gw_gw_03_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_gw_gw_03_3_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_gw_gw_03_solution_2&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bfseries&#34;&gt;Predicted Earth Temperature:&lt;/b&gt; &lt;/p&gt; &lt;p&gt; If you did your calculation correctly, your value for [mathjaxinline]T_ E[/mathjaxinline] is &lt;b class=&#34;bfseries&#34;&gt;too low&lt;/b&gt; compared to the actual observed average temperature of the surface of Earth, 288 K, which is equal to 15[mathjaxinline]^ o[/mathjaxinline]C. Our model was very simple and did not consider the fact that the atmosphere does abosrb the infrared radiation of Earth. In the next problem we will add a single layer of atmosphere to the Bare Rock model and evaluate its effect on the predicted temperature. &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bfseries&#34;&gt;Bare Rock Model Assumptions:&lt;/b&gt; &lt;/p&gt; &lt;p&gt; Below we summarize the assumptions made in the Bare Rock Model &lt;/p&gt; &lt;ol class=&#34;enumerate&#34;&gt; &lt;li value=&#34;1&#34;&gt; &lt;p&gt; The atmosphere transmits all the solar radiation (it is transparent in wavelengths characteristic of the sun&#39;s radiation). &lt;/p&gt; &lt;/li&gt; &lt;li value=&#34;2&#34;&gt; &lt;p&gt; The Earth behaves as a perfect spherical blackbody at constant temperature [mathjaxinline]T_ E[/mathjaxinline]. It absorbs all the incoming solar radiation and it emits infrared radiation following the Stefan-Boltzmann law: [mathjaxinline]I_{out} = \sigma T_ E^4[/mathjaxinline] &lt;/p&gt; &lt;/li&gt; &lt;li value=&#34;3&#34;&gt; &lt;p&gt; The atmosphere transmits all of Earth&#39;s radiation. (It is transparent to Earth&#39;s infrared radiation, which is longer in wavelength than the wavelength of visible light.) &lt;/p&gt; &lt;/li&gt; &lt;/ol&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 3: The Bare Rock Model &#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_gw_gw_03&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_gw_gw_03&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="False" data-init="VerticalStudentView" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-global_warming_5" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="836e39be05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">Problem 4</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_04"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_04" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="836e39be05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_gw_gw_04" class="problems-wrapper" role="group" aria-labelledby="gw_gw_04-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_04" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_04/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="4" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;gw_gw_04-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_04-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 4: One Layer Model &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@gw_gw_04-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; The Bare Rock Model predicts too low a temperature because it does not consider the absorption of radiation from the atmosphere. The greenhouse effect is the result of the absorption of the infrared radiation emitted by the surface of Earth by the gases in the atmosphere (water vapor, carbon dioxide, and methane). We can make a simple model of the impact that this absorption has on the temperature of teh surface Earth by modeling the atmosphere as a single layer with the following assumptions: &lt;/p&gt; &lt;ol class=&#34;enumerate&#34;&gt; &lt;li value=&#34;1&#34;&gt; &lt;p&gt; The atmosphere is transparent to the incoming solar energy. &lt;/p&gt; &lt;/li&gt; &lt;li value=&#34;2&#34;&gt; &lt;p&gt; The surface of Earth radiates as a perfect spherical blackbody at a temperature [mathjaxinline]T_ E[/mathjaxinline]. &lt;/p&gt; &lt;/li&gt; &lt;li value=&#34;3&#34;&gt; &lt;p&gt; The single layer atmosphere absorbs all the infrared radiation emitted by Earth. &lt;/p&gt; &lt;/li&gt; &lt;li value=&#34;4&#34;&gt; &lt;p&gt; The atmosphere then radiates as a perfect blackbody at [mathjaxinline]T_ A[/mathjaxinline], with an outward radiation away from the surface of Earth and an inward radiation towards the surface of Earth. &lt;/p&gt; &lt;/li&gt; &lt;li value=&#34;5&#34;&gt; &lt;p&gt; Radiative equilibrium is the condition that the incoming solar radiation is equal to the outward radiation from the atmosphere. &lt;/p&gt; &lt;/li&gt; &lt;li value=&#34;6&#34;&gt; &lt;p&gt; Radiative equilibrium on the surface of Earth is the condition that the incoming solar radiation and the downward atmospheric radiation is equal to the radiation from the surface. &lt;/p&gt; &lt;/li&gt; &lt;/ol&gt; &lt;p&gt; The different types of radiation involved in the Single Layer Model are indicated in the energy diagram below. Note that the single layer of atmosphere emits radiation downward from its bottom surface and radiation upward back to space from its top surface. &lt;/p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/0ebf43e03a3d5235d0d096e72451f36e/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_gw_04_b.svg&#34; width=&#34;550&#34;/&gt; &lt;/center&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; Consider the single layer of atmosphere to be at a constant temperature [mathjaxinline]T_ A[/mathjaxinline]. Apply the radiative equilibrium condition to the space side of the layer of atmosphere to obtain an expression for [mathjaxinline]T_ A^4[/mathjaxinline]. Express your answer in terms of alpha for [mathjaxinline]\alpha[/mathjaxinline], sigma for [mathjaxinline]\sigma[/mathjaxinline], S_0 for [mathjaxinline]S_0[/mathjaxinline], and T_E for [mathjaxinline]T_ E[/mathjaxinline] as needed. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]T_ A^4=[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_gw_gw_04_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_gw_gw_04_2_1&#34; id=&#34;input_gw_gw_04_2_1&#34; aria-describedby=&#34;status_gw_gw_04_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_gw_gw_04_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_gw_gw_04_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_gw_gw_04_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_gw_gw_04_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_gw_gw_04_2_1_dynamath&#34; name=&#34;input_gw_gw_04_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_gw_gw_04_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part b)&lt;/b&gt; Consider the surface of Earth to be at a constant temperature [mathjaxinline]T_ E[/mathjaxinline]. Apply the radiative equilibrium condition to the surface of Earth to obtain an expression for [mathjaxinline]T_ E^4[/mathjaxinline]. Express your answer in terms of T_A for [mathjaxinline]T_ A[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]T_ E^4=[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_gw_gw_04_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_gw_gw_04_3_1&#34; id=&#34;input_gw_gw_04_3_1&#34; aria-describedby=&#34;status_gw_gw_04_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_gw_gw_04_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_gw_gw_04_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_gw_gw_04_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_gw_gw_04_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_gw_gw_04_3_1_dynamath&#34; name=&#34;input_gw_gw_04_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_gw_gw_04_solution_2&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part c)&lt;/b&gt; Define [mathjaxinline]\displaystyle T_{BRM}[/mathjaxinline] to be the Earth&#39;s temperature predicted by the Bare-Rock-Model you calculated in the previous problem, [mathjaxinline]\displaystyle T_{BRM}=(\frac{1-\alpha }{4\sigma }S_0)^{1/4}[/mathjaxinline]. What is the ratio [mathjaxinline]T_ E/T_{BRM}[/mathjaxinline]? &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_gw_gw_04_4_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_gw_gw_04_4_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_gw_gw_04_4_1&#34; id=&#34;input_gw_gw_04_4_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;gw_gw_04_4_1-choice_1-label&#34; for=&#34;input_gw_gw_04_4_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_gw_gw_04_4_1&#34;&gt; &lt;text&gt; [mathjaxinline]T_ E/T_{BRM} =2[/mathjaxinline]&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_gw_gw_04_4_1&#34; id=&#34;input_gw_gw_04_4_1_choice_2&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;gw_gw_04_4_1-choice_2-label&#34; for=&#34;input_gw_gw_04_4_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_gw_gw_04_4_1&#34;&gt; &lt;text&gt; [mathjaxinline]T_ E/T_{BRM} =1/2[/mathjaxinline]&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_gw_gw_04_4_1&#34; id=&#34;input_gw_gw_04_4_1_choice_3&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;gw_gw_04_4_1-choice_3-label&#34; for=&#34;input_gw_gw_04_4_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_gw_gw_04_4_1&#34;&gt; &lt;text&gt; [mathjaxinline]T_ E/T_{BRM} =(2)^{1/4}[/mathjaxinline]&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_gw_gw_04_4_1&#34; id=&#34;input_gw_gw_04_4_1_choice_4&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_4&#34;/&gt;&lt;label id=&#34;gw_gw_04_4_1-choice_4-label&#34; for=&#34;input_gw_gw_04_4_1_choice_4&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_gw_gw_04_4_1&#34;&gt; &lt;text&gt; [mathjaxinline]T_ E/T_{BRM} =1/(2)^{1/4}[/mathjaxinline]&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_gw_gw_04_4_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_gw_gw_04_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_gw_gw_04_solution_3&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part d)&lt;/b&gt; Calculate the temperature of the surface of Earth predicted by the Single Layer model. Enter your answer in K. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]T_ E=[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_gw_gw_04_5_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_gw_gw_04_5_1&#34; id=&#34;input_gw_gw_04_5_1&#34; aria-describedby=&#34;trailing_text_gw_gw_04_5_1 status_gw_gw_04_5_1&#34; value=&#34;&#34; size=&#34;10&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_gw_gw_04_5_1&#34;&gt;[mathjaxinline]\mathrm{(K)}[/mathjaxinline]&lt;/span&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_gw_gw_04_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_gw_gw_04_5_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_gw_gw_04_solution_4&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bfseries&#34;&gt;Conclusions:&lt;/b&gt; &lt;/p&gt; &lt;p&gt; The temperature that you calculated is too high. This is due to overly simplifying assumptions that we have made about the atmosphere. The next step is to consider a more dynamic model of the atmosphere that includes the study of radaitive-convective equilibrium, different rates of absorption of radiation by the different gasses, and a host of other effects that requires a more detailed understanding of atmospheric physics. &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 4: One Layer Model &#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_gw_gw_04&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_gw_gw_04&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. 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